In order to develop the distance ladder from
local to distant cosmic objects, one starts from the local distance scale
(for a detailed account see
[143],
[144]).

Galactic Distance Indicators: The primary method used to
estimate the distances to nearby stars is that of Trigonometric
Parallax.
As the Earth orbits the Sun, we view the Universe from different points
of the orbit throughout the year. This difference is maximum every 6
months when the Earth is on the opposite side of its orbit around the
Sun. This kinematic method provides the basic unit of distance, the
parsec, used by astronomers. It is defined as the distance at
which a star would have a trigonometric parallax of 1"
(parsec = parallax + second):

Note that this method is effective out to ~ 60 parsecs and that the
nearest star to us
(-Centauri) is at a
distance of 0.75 parsecs.

Among the many distance indicators, used to determine distances
within our Galaxy, a particularly interesting one is the Main
sequence fitting method. This takes advantage of the fact that stars
in globular
clusters are at a common distance and that there is a unique correlation
between spectral stellar type and absolute luminosity (the H-R
diagram).
Therefore by measuring the distance, via a kinematic method, to one
relatively near globular cluster, one sets the zero-point of this method
and then by observing the apparent magnitude - spectral type
distribution of other globular clusters, one can determine their
distance.

Extragalactic Distance Indicators: The next step is based on
Cepheid Variable Stars. This method has been traditionally used
within our Galaxy and in the nearby Large Magellanic Cloud (LMC), but
with the Hubble Space Telescope it has been successfully used
out to ~ 20 Mpc (cf.
[61]).
A strong and tight relationship exists
between the intrinsic luminosity of these stars and their period of
variation (pulsation) which results in a Period-Luminosity relation:

(51)

Once this relation has been calibrated, it provides the absolute
luminosity of the distant Cepheid stars and via (39), the distance of their
host galaxy. Although this relation has a scatter, in the I-band,
of only
±0.1 mag, systematic effects may exist. For example, a
serious concern is
whether there is any environmental dependence of the relation. It has
been suggested that a different metalicity of the host galaxy may
significantly affect the zero-point of the relation and thus the determined
distance. These effects can be taken into account and this method has
proved to be fundamental in the recent determinations of
H0, because it
provides the link between the primary galactic indicators and the
local
extragalactic ones, which then provide the calibration for other
secondary indicators operating in much larger distances (cf.
[61]
and references therein).

In developing the distance scale, we now need effective indicators that
can be used to very large distances. Other scaling relations have been
found between a distance dependent (ex. brightness, diameter) and a
distance independent (ex. rotational velocity, stellar velocity dispersion)
quantity. It is evident that from such relations one can extract distance
information. The main assumption in such a use of these scaling relations
is that they are not environment-dependent (which has been shown to be
a valid assumption). Such a relation for spiral galaxies is the
Tully-Fisher relation
[175],
which relates the rotational velocity of a spiral to its total
infrared luminosity: LirVrot4 or its total blue luminosity:
LbVrot with
~ 2.4 -2.8. It has a reasonable theoretical justification:
Rotational velocities in spirals are related to mass according to

(52)

Assuming that all spirals have the same surface brightness S, then

(53)

If the mass to light ratio is constant then

(54)

For ellipticals a similar relation holds, the Faber-Jackson
relation
[60],
which relates the absolute luminosity of the galaxy with the stellar
velocity dispersion (L3-4) or a
variant, the so-called Dn -
relation
(cf. [50]),
which relates the diameter Dn of an elliptical
(defined as that
within which the mean surface brightness is 20.75 mag arc
sec-2 in B) to
the stellar velocity dispersion
:
Dnx
with x ~ 1.2 -1.3 .
The typical accuracy of these distance estimations is ~ 20% and
the usual assumption is that they do not evolve with redshift
over the scales used, and that systematic effects can be corrected
for. However, there are some indications for an evolution of the
B-band Tully-Fisher relation in the interval studied 0.1
< z < 1
[195].

Another very important distance indicator, which can be used to
large distances, is the Supernova SNIa brightness at maximum. It is
thought that the maximum luminosity of such supernovae (explosion of
white dwarfs in binary systems which become gravitationally unstable -
reaching the Chandrasekhar limit - due to the accretion of matter from
the secondary - see
[92])
is a Universal constant and since the intrinsic
luminosity of a SNIa is high, they can be seen out to cosmological
distances. Furthermore, a correlation was found between the supernova
luminosity and the brightness decay time, which provides a further
parameter that reduces the scatter in luminosity to ±0.3 mags. Using
this distance indicator one can construct the Hubble diagram to very
large-distances and thus determine the deceleration of the Universe by
mapping the region of the Hubble diagram that deviates from
linearity. This has been recently achieved by two independent groups,
the Supernova Cosmology Project - SCP
[118]
and the High-zSupernova Search Team - HZT
[156],
which have found that the derived
Hubble diagram is that expected from a accelerating expansion,
which can be provided by a non-zero cosmological constant (see
Fig.5).

Although I do not plan to present all the secondary distance
indicators, one that is potentially very important and susceptible to
small uncertainties and systematics, is the surface brightness
fluctuation
method. This method is based on the fact that the discreteness of stars
within galaxies depends on distance. This methods has an accuracy of
~ 5% in distance.

Direct Distance Indicators: Clusters of
galaxies are filled
with hot and transparent gas which can be fitted by a thermal
Bremsstrahlung spectrum with t ~ 5 × 107
-108k (covering a range
of 2 ~ 8 KeV). The physics of the hot intercluster gas provides
the means of measuring directly the distance of clusters, without
need of intermediate steps. This method is based on the so called
Sunyaev-Zeldovich effect which is the distortion of the original CMB
spectrum, by the Compton scattering to higher energies of the CMB
photons from the hot electrons of the plasma (for recent reviews see
[131],
[17]).
This distortion decreases the brightness of the CMB
spectrum at the longer wavelength range while it increases the photon
energies in the shorter wavelength range. Lets assume a cluster of radius
R at a distance D, subtending an angular separation
on the sky. The
change of the CMB brightness temperature is proportional to
the line integral of electron number density through the cluster:

(55)

then from isothermality we get
T/TTeneR. Furthermore the
luminosity of the Bremsstrahlung radiation together with the assumption
of isothermality, gives:

(56)

From (56), the observed X-ray flux (FxLx / D2) and R =
D we have:

(57)

and solving for D we get:

Therefore measuring Fx,
,
Te we obtain an absolute determination of
the cluster distance. The temperature, Te, can be
measured either from
the shape of the X-ray continuum or from line emission (especially of
iron). Note however, that we have assumed sphericity, isothermality and
a smooth distribution of ne. Most clusters are
flattened (cf.
[12]),
and show significant substructure apparent in the optical and X-ray images.
In such clusters the above procedure may provide highly uncertain
and biased distance estimates (cf.
[77]).
A recent study of ~ 100
clusters has shown that once cooling flows are taken into account,
isothermal profiles fit well ~ 90% of the clusters
([185],
but see also
[45]).